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6.1: The Mean

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

• Understand the mean of a set of numerical data.
• Compute the mean of a given set of data.
• Understand the mean of a set of data as it applies to real world situations.

Introduction

You are getting ready to begin a unit in Math that deals with measurement. Your teacher wants you to use benchmarks to measure the length of some objects in your classroom. A benchmark is simply a standard by which something can be measured. One of the benchmarks that you can all use is your hand span. Every student in the class must spread their hand out as far as possible and places it on top of a ruler or measuring tape. The distance from the tip of your thumb to the tip of your pinky is your hand span. Your teacher will record all of the measurements. The following results were recorded by a class of thirty-five students:

Hand span (inches) Frequency
\begin{align*}6 \frac{1}{2}\end{align*} 1
\begin{align*}7 \frac{1}{4}\end{align*} 3
\begin{align*}7 \frac{1}{2}\end{align*} 8
\begin{align*}7 \frac{3}{4}\end{align*} 10
\begin{align*}8 \frac{1}{4}\end{align*} 7
\begin{align*}8 \frac{1}{2}\end{align*} 4
\begin{align*}9 \frac{1}{4}\end{align*} 2

Later in this lesson, we will compute the mean or average hand span for the class.

The term “central tendency” refers to the middle value or a typical value of the set of data which is most commonly measured by using the three m’s – mean, median and mode. In this lesson we will explore the mean and then move onto the median and the mode in the following lessons.

The mean, often called the ‘average’ of a numerical set of data, is simply the sum of the data numbers divided by the number of numbers. This value is referred to as an arithmetic mean. The mean is the balance point of a distribution.

Example 1: In a recent hockey tournament, the number of goals scored by your school team during the eight games of the tournament were 4,5,7,2,1,3,6,4. What is the mean of the goals scored by your team?

Solution: You are really trying to find out how many goals the team scored each game.

• The first step is to add the number of goals scored during the tournament.

\begin{align*}4 + 5+ 7 + 2 + 1 + 3 + 6 + 4 = 32 \ (\text{The sum of the goals is} \ 32)\end{align*}

• The second step is to divide the sum by the number of games played.

\begin{align*}32 \div 8 = 4\end{align*}

From the calculations, you can say that the team scored a mean of 4 goals per game.

Example 2: The following numbers represent the number of days that 12 students bought lunch in the school cafeteria over the past two months. What is the mean number of times that each student bought lunch at the cafeteria during the past two months?

\begin{align*}22, 23, 23, 23, 24, 24, 25, 25, 26, 26, 29, 30\end{align*}

Solution: The mean is \begin{align*}\frac{22 + 23 + 23 + 23 + 24 + 24 + 25 + 25 + 26 + 26 + 29 + 30}{12}\end{align*}

The mean is \begin{align*}\frac{300}{12}\end{align*}

The mean is 25

Each student bought lunch an average of 25 times over the past two months.

If we let \begin{align*}x\end{align*} represent the data numbers and \begin{align*}n\end{align*} represent the number of numbers, we can write a formula that can be used to calculate the mean \begin{align*}\bar{x}\end{align*} of the data. The symbol \begin{align*}\sum\end{align*} means ‘the sum of’ and can be used when we write a formula for calculating the mean.

\begin{align*}\bar{x} = \frac{\sum x_1 + x_2 + x_3 + \ldots + x_n}{n}\end{align*}

If we are given a large number of values and if some of them appear more than once, the data is often presented in a frequency table. This table will consist of two columns. One column will contain the data and the second column will indicate the how often the data appears. Although the data given in the above problem is not large, some of the values do appear more than once. Let’s set up a table of values and their respective frequencies as follows:

Number of Lunches Bought Number of Students
22 1
23 3
24 2
25 2
26 2
29 1
30 1

Now, the mean can be calculated by multiplying each value by its frequency, adding these results, and then dividing by the total number of values (the sum of the frequencies). The formula that was written before can now be written to accommodate the values that appeared more than once.

\begin{align*}\bar{x} = \frac{\sum x_1 f_1 + x_2 f_2 + x_3 f_3 + \ldots + x_nf_n}{f_1 + f_2 + f_3 + \ldots + f_n}\end{align*}

We see that this answer agrees with the result of Example 2.

Besides doing these calculations manually, you can also use the TI83 calculator. Example 2 will be done using both methods and the TI83.

Step One:

Step Two:

Notice the sum of the data \begin{align*}\left (\sum x \right ) = 300\end{align*}

Notice the number of data \begin{align*}(n=12)\end{align*}

Notice the mean of the data \begin{align*}(\bar{x}=25)\end{align*}

Example 2 was done using the TI83 calculator by using List One only. Now we will do Example 2 again but this time we will utilize the TI83 as a frequency table.

Step One:

Step Two:

Step Three:

Step Four:

Press \begin{align*}2^{nd}\end{align*} 0 to obtain the CATALOGUE function of the calculator. Scroll down to sum( and enter \begin{align*}L_3 \ \rightarrow\end{align*}

You can repeat this step to determine the sum of \begin{align*}L_2 \ \rightarrow\end{align*}

\begin{align*}\bar{x} = \frac{300}{12} = 25\end{align*}

A frequency table can also be drawn to include a tally column. To calculate the mean of a set of data, the values do not have to be arranged in ascending (or descending order). Therefore, the tally column acts as a speedy method of determining the frequency of each value.

Example 3: A survey of 30 students with cell phones was conducted by teachers to determine the mean number of hours a student spends each week on their cell phone.

Following are the estimated times:

12, 15, 20, 8, 25, 11, 8, 11, 15, 14, 14, 20, 18, 13, 8, 28, 12, 12, 13, 20, 5, 8, 13, 11, 5, 18, 24, 16, 14, 18

Time (Hours) Tally Number of Students
12 ∕ ∕ ∕ 3
15 ∕ ∕ 2
20 ∕ ∕ ∕ 3
8 ∕ ∕ ∕ ∕ 4
25 ∕ ∕ 1
11 ∕ ∕ ∕ 3
14 ∕ ∕ ∕ 3
18 ∕ ∕ ∕ 3
13 ∕ ∕ ∕ 3
28 1
5 ∕ ∕ 2
24 1
16 1

Now that the frequency for each value has been determined the mean can now be calculated:

Solution:

\begin{align*}\bar{x} & = \frac{\sum x_1 f_1 + x_2 f_2 + x_3 f_3 + \ldots + x_n f_n}{f_1 + f_2 + f_3 + \ldots + f_n} \\ \bar{x} & = \frac{12(3) + 15(2) + 20(3) + 8(4) + 25 + 11(3) + 14(3) + 18(3) + 13(3) + 28 + 5(2) + 24 + 16}{3+2+3+4+1+3+3+3+3+1+2+1+1} \\ \bar{x} & = \frac{429}{30} \\ \bar{x} & = 14.3\end{align*}

The mean amount of time that each student spends using a cell phone is 14.3 hours.

Now we will return to the problem that was posed at the beginning of the lesson – the one that dealt with hand spans.

Hand span (inches) Frequency
\begin{align*}6 \frac{1}{2}\end{align*} 1
\begin{align*}7 \frac{1}{4}\end{align*} 3
\begin{align*}7 \frac{1}{2}\end{align*} 8
\begin{align*}7 \frac{3}{4}\end{align*} 10
\begin{align*}8 \frac{1}{4}\end{align*} 7
\begin{align*}8 \frac{1}{2}\end{align*} 4
\begin{align*}9 \frac{1}{4}\end{align*} 2

Solution:

\begin{align*}\bar{x} & = \frac{\sum x_1 f_1 + x_2 f_2 + x_3 f_3 + \ldots + x_n f_n}{f_1 + f_2 + f_3 + \ldots + f_n} \\ \bar{x} & = \frac{6 \frac{1}{2} + 7\frac{1}{4}(3) + 7\frac{1}{2}(8) + 7\frac{3}{4}(10) + 8\frac{1}{4}(7) + 8 \frac{1}{2}(4) + 9\frac{1}{4}(2)}{1 + 3 + 8 + 10 + 7 + 4 + 2} \\ \bar{x} & = \frac{276}{35} \\ \bar{x} & = 7\frac{31}{35} \approx 7.89\end{align*}

The mean hand span for the 35 students is approximately 7.89 inches.

Lesson Summary

You have learned the significance of the mean as it applies to a set of numerical data. You have also learned how to calculate the mean when the data is presented as a list of numbers as well as when it is represented in a frequency table. To facilitate the process of calculating the mean, you have also learned to apply the formulas necessary to do the calculations.

Points to Consider

• Is the mean only important as a measure of central tendency?
• If data is represented in another way, is it possible to either calculate or estimate the mean from this other representation?

Review Questions

1. Find the mean of each of the following sets of numbers:
1. 3, 5, 5, 7, 4, 8, 6, 2, 5, 9
2. 8, 3, 2, 0, 4, 3, 4, 6, 7, 9, 5
3. 3, 8, 4, 1, 8, 7, 5, 6, 3, 7, 2, 9
4. 18, 28, 27, 27, 23, 22, 25, 21, 1
2. The number of days it rained during four months were: April – 11 days May – 8 days June – 13 days July – 24 days Find the mean number of rainy days per month.
3. Busy Bobby earned the following amounts of money over a four week period: Week One - \begin{align*}\106.64\end{align*} Week Two - \begin{align*}\120.42\end{align*} Week Three - \begin{align*}\110.54\end{align*} Week Four - \begin{align*}\122.16\end{align*} Find the mean weekly wage.
4. Mary Hop must ride to her workplace on the bus. She found that the number of minutes she spent riding on the bus each day was different. \begin{align*}|\end{align*}Following are the number of minutes she recorded for the five work days last week: Monday – 43 minutes Tuesday – 50 minutes Wednesday – 47 minutes Thursday – 49 minutes Friday – 41 minutes How many minutes are there in the mean trip?
5. The number of fans that attended the last six games of the local baseball team during the cup competition were: \begin{align*}5200, 8130, 11250, 13208, 18750, 24060\end{align*} What was the mean attendance for each game?
6. Two dice were thrown together six times and the results are shown below: First Throw – 3 Second Throw – 7 Third Throw – 11 Fourth Throw – 9 Fifth Throw – 12 Sixth Throw – 6 What is the mean of these throws of the dice?
7. The frequency table below shows the number of Tails when four coins are tossed 64 times. What is the mean?
Number Of Tails Frequency
4 3
3 23
2 16
1 17
0 5
1. A manufacturer of light bulbs had their quality control department test the lifespan of their bulbs. Forty-two bulbs were randomly selected and tested, with the number of hours they lasted listed below. \begin{align*}& 100 \quad 125 \quad 137 \quad 167 \quad 158 \quad 110 \quad 142 \\ & 163 \quad 135 \quad 146 \quad 134 \quad 121 \quad 163 \quad 168 \\ & 114 \quad 128 \quad 164 \quad 152 \quad 158 \quad 143 \quad 162 \\ & 137 \quad 126 \quad 149 \quad 168 \quad 152 \quad 129 \quad 156 \\ & 153 \quad 162 \quad 168 \quad 144 \quad 124 \quad 119 \quad 147 \\ & 147 \quad 152 \quad 162 \quad 159 \quad 157 \quad 141 \quad 160\end{align*} If the manufacturer wants to offer a warranty with the light bulbs, what is the mean number of hours that the bulbs lasted?
2. The following data represents the height in centimeters of 32 Grade 10 students. What is the mean height of the students? \begin{align*}& 158 \quad 169 \quad 156 \quad 174 \quad 180 \quad 163 \quad 162 \quad 159 \\ & 167 \quad 179 \quad 181 \quad 167 \quad 170 \quad 164 \quad 172 \quad 175 \\ & 161 \quad 174 \quad 176 \quad 182 \quad 173 \quad 168 \quad 160 \quad 183 \\ & 157 \quad 165 \quad 174 \quad 169 \quad 180 \quad 176 \quad 168 \quad 180\end{align*}
3. Miss Smith gave her class a surprise quiz and gave it a value of 15 points. The following frequency table shows the results:
Quiz Mark Number of Students
0 0
1 0
2 0
3 0
4 1
5 2
6 2
7 4
8 5
9 6
10 3
11 4
12 1
13 0
14 1
15 0

What was the mean mark scored by the class?

1. A traveling salesman buys gasoline for his car every day. The table below shows the number of gallons of gasoline he bought each day over a span of 42 days.
Number of Gallons Number of Days
2 6
3 9
4 5
5 14
6 8

Find the mean number of gallons of gasoline he bought each day.

1. When four dice were thrown together a total of 200 times, the number of threes scored per throw is shown in the table. Calculate the mean number of threes scored each throw.
Number of 3’s Number of Throws
4 1
3 2
2 13
1 34
0 150
1. The table below shows the number of touchdowns scored by a football team during each of 50 games. Determine the number of touchdowns the team scored each game.
Number of Touchdowns Number of Games
6 1
5 2
4 4
3 8
2 10
1 12
0 13
1. My Grade 11 Math class has thirty-two students. The following table shows the frequency of attendance over a period of 30 days. Find the mean daily attendance.
Number of Students Present Number of Days
25 1
26 1
27 1
28 2
29 8
30 7
31 6
32 4
1. The following table shows the number of passengers that used the Handi-Trans bus over a period of 60 days. Calculate the mean number of passengers on the bus each day.
Number of Passengers Number of Days
3 16
4 12
5 10
6 7
7 8
8 7

1. (5.4)
2. (4.64)
3. (5.25)
4. (21.33)
1. 14 rainy days
2. \begin{align*}(\114.94)\end{align*}
3. 46 minutes
4. (13433 fans)
5. 8
6. (2.03)
7. 145.29 hours
8. (169.97 cm.)
9. 8.55
10. (4.21 gallons daily)
11. 0.35 threes
12. (1.76 touchdowns)
13. 29.67 students
14. (5 passengers)

Answer Key for Review Questions (even numbers)

2. 14 rainy days

4. 46 minutes

6. 8

8. 145.29 hours

10. 8.55

12. 0.35 threes

14. 29.67 students

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